Method and apparatus for constructing images from measurements of impedence

ABSTRACT

Apparatus and methods are disclosed for obtaining an image of the impedance properties of an object by inferring a measure of straight-line path impedance along a plurality of paths from a plurality of current amplitude and phase measurements made between combinations of electrodes placed at selected points, and/or from measurements of the intensity of the electromagnetic signal emitted when an alternating current is made to resonate along the straight line path.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of and claims the benefit of U.S. Ser.No. 11/738,404, filed Apr. 20, 2007 now U.S. Pat. No. 8,026,731,allowed, which claims the benefit of U.S. Provisional Application No.60/794,219 filed Apr. 20, 2006. The disclosures of each of the foregoingare incorporated by reference herein.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH & DEVELOPMENT

Not applicable

BACKGROUND—DESCRIPTION OF PRIOR ART

Many investigators have attempted, to devise methods for producingimages corresponding to the electrical impedance properties of a two orthree dimensional object based upon measurements taken from electrodesplaced around the outside of (and/or, optionally, at selected pointswithin) the object. Potential applications of such a technology(referred to as electrical impedance tomography, or ‘EIT’) includemedical imaging and diagnosis, and geological profiling.

Prior art approaches to the problem have typically involved passingcurrents between various pairs of electrodes arrayed around theperiphery of the object to be imaged. Multiple frequencies may beemployed, and/or currents may be passed through multiple electrodes atonce. Thus far, however, there has been relatively little success inreconstructing meaningful and reproducible images at a usefulresolution, because of a seemingly intractable problem: currents appliedthrough electrodes follow multiple paths of least resistance thatthemselves depend on the impedance characteristics of the object and aretherefore unpredictable. Because of this, impedance imaging is inprinciple unlike other kinds of imaging such as computed tomography,where the measured signal represents an integral of the property beingmeasured (typically density to x-rays) along a straight line path, whichallows for straightforward reconstruction of a unique image from anumber of such measurements along a variety of paths. In computedtomography where the measured data consists of line integrals of somephysical property, reconstruction is accomplished using techniques thatarc well known to persons having ordinary skill in the art of imaging,such as back projection or Fourier analysis. Because the measurementssought to be used for impedance imaging are not line integrals, butrather represent the effect of the impedance properties of the object tobe imaged along many paths at once, the problem of reconstructing animage from impedance measurements (often referred to as the ‘inverse EITproblem’) is one of a class of inverse problems known to be highlynonlinear, extremely ill-posed, and having many local optima.

Prior art efforts to obtain useful images despite these drawbacks havetypically focused on seeking ways to make the reconstruction problemless intractable—for example, in physiological imaging, the analysis maybegin with an assumed mapping of the typical impedance properties andtopography of the anatomical region sought to be measured. It may thenbe possible to construct an image at some resolution by using theassumed mapping to predict the path distribution of the appliedcurrents, and use the results of the measurements to iteratively improvethe mapping.

The present invention takes a different approach: it seeks to makemeasurements in such a way that the current along a straight line pathbetween electrodes can be estimated, thereby providing line integralsfrom which images can be reconstructed directly using any of the manywell-known line integral-based image reconstruction methods.

SUMMARY OF THE INVENTION

The present invention offers an improved system and method (andcomputing apparatus and measurement apparatus implementing such method)for obtaining an image of the impedance properties of an object from aplurality of current measurements made between combinations ofelectrodes placed at selected points, typically around the periphery ofthe object but also optionally within the interior of the object. Themeasurements in question are able to estimate a measure of the impedancealong a straight line path between electrodes by taking advantage ofeither or both of two novel improvements, described in detail infra: (1)the use of the phase shifts and current amplitudes measured at aplurality of frequencies, together with knowledge of the distancebetween electrodes, to determine the part of the current attenuationattributable to the straight line path, and/or (2) the use of acalibrated frequency to produce a standing wave along the straight linepath, thereby, in effect, interrogating the straight line path with asignal that is resonant along that path.

BRIEF DESCRIPTION OF THE DRAWINGS

Reference will be made to the accompanying drawings, which illustratenon-limiting examples, embodiments, and/or aspects of the invention.

FIG. 1 illustrates the modeling of impedance properties of an object forcomputational purposes.

FIG. 2 illustrates the modeling of impedance properties of an object forcomputational purposes taking into account capacitance.

FIG. 3 illustrates the modeling of impedance properties of a region by asingle lumped impedance.

FIG. 4 illustrates a method for measuring impedance related propertiesof a conductive path through an object upon application of a signalresonating on the desired path.

FIG. 5 illustrates an apparatus for measuring impedance relatedproperties of an object.

DETAILED DESCRIPTION OF THE INVENTION

The motivation of the method and apparatus described here is to solvethe problem of the fundamental intractability of the impedance imagingproblem by obtaining measurements from which line integrals, or at leastbounded estimates of line integrals, of impedance properties can beobtained. Current passed between electrodes follows multiple paths, andthere is no known reliable technique for physically confining theelectrical current to a predetermined path, so in order to obtain lineintegral measurements, it is necessary to invent a way of determiningwhich part of the injected current did in fact follow the straight linepath. The discussion to follow will describe a system and method formaking that determination, involving several related techniques that canbe used separately or together and that enable the present invention.

Determination of Line Integral Impedance Properties from Amplitude andPhase Shift Measurements at Multiple Frequencies

Consider an alternating current signal of predetermined frequency 30applied between two electrodes placed on the periphery of an object tobe imaged 21, 22, as shown in FIG. 1. The resistance properties of theobject can be approximated by an array of parallel resistors R₁ 26, R₂27, and R₃ 28 as shown, each on a separate circuit branch 23, 24, 25respectively, which branches are of length D₁, D₂, and D₃ respectively.(The approximation can be improved using a larger number of parallelpaths: in this illustration only three are shown, for simplicity. Thesystem and method described is generalizable to any number of paths.Capacitance and inductance effects are also ignored in this example forsimplicity, as are resonant effects; these will be discussed infra.) Ifa sinusoidal signal at a fixed frequency f and specified potential V(herein referred to as an “interrogation signal”) is applied across theelectrodes 21 and 22, alternating currents will be produced in each ofthe circuit branches 23, 24, 25, and these alternating currents willsuperpose at the ‘downstream’ electrode 22, producing a compositecurrent signal, still of frequency f but, in general, shifted in phaseand attenuated in amplitude from the ‘upstream’ current at the electrode21. This phase shift and attenuation occurs because all branches exceptthe straight line path 23 are longer than the straight line path, so thesignals on the non-straight line paths (hereinafter referred to as theshunt paths) are relatively delayed in their arrival at the ‘downstream’electrode 22 by an amount of time that varies according to the ratio ofthe shunt path length to the straight line path length. Thus when thesignals from the several branches superpose at the ‘downstream’electrode, they interfere destructively to a greater or lesser degree,causing the downstream current to be attenuated and phase shifted. (Thisresult may seem counterintuitive, since the usual principles of circuitanalysis would require the ‘upstream’ current to equal the ‘downstream’current. However, the usual principles of circuit analysis do not takeinto account propagation delay due to the differences in path length,since the attenuation and phase shift is negligible at the frequenciesusually analyzed, which correspond to wavelengths that are largerelative to the path lengths. Note: as used herein, two or more signalsare said to interfere constructively if and only if they are in phase,so that the amplitude of the signal obtained by superposing them isequal to the sum of the amplitudes of the separate, signals. They aresaid to interfere destructively if any of the signals being superposedis out of phase to any significant degree, in which case the amplitudeof the signal obtained by superposing them will always be less than thesum of the amplitudes of the separate signals.)

The attenuation and phase shift caused by the interference of thesignals traveling along the various paths can be used to infer theamplitude of the individual path currents, in the manner disclosedherein. In general, doing so requires knowledge of (1) the fixedpotential of each applied interrogation signal, (2) the lengths of eachof the current paths, and (3) the amplitude of the current at the‘downstream’ electrode and the phase of that current relative to theapplied signal or some other datum, with measurements taken at a numberof fixed frequencies at least equal to the number of paths to beevaluated. The discussion to follow will first describe how this may bedone in the context of the simplified model of FIG. 1, then discuss howthe model may be generalized to the problem of estimating the impedancealong straight line paths through two or three dimensional objects.

Consider an alternating current signal of fixed potential V andfrequency f₁ applied across electrodes 21 and 22 in FIG. 1. A currentI_(1,1) will be produced in the straight line path 23 equal to V/R₁ andwill be phase shifted relative to the applied signal by an angle inradians equal to 2π(D₁−λ₁)/λ₁, where λ₁ is the wavelength correspondingto f₁ and given by λ₁=c/f₁, where c the speed of light in the mediumthrough which the current is passing. A current I_(1,2) equal to V/R₂will be produced in the shunt path 24, and will be phase shiftedrelative to I₁ by an amount equal to 2π(D₂−λ₁)/λ₁. Similarly, A currentI_(1,3) equal to V/R₃ will be produced in the shunt path 25, and will bephase shifted relative to I₁ by an amount equal to 2π(D₃−λ₁)/λ₁. (Forconsistency, a current I_(x,y) will be taken throughout this writtendescription to mean the current in branch y due to application offrequency f_(x). A current I_(x) will be taken to refer to the aggregatesuperposed current from all paths measured at the downstream electrodedue to application of frequency f_(x).)

It is convenient to represent these currents I_(1,1), I_(1,2), I_(1,3)in complex form as (V/R₁) cos(2π(D₁−λ₁/λ₁)+j((V/R₁) sin(2π(D₂−λ₁/λ₁)),(V/R₂) cos(2π(D₂−λ₁/λ₁)+j((V/R₂) sin(2π(D₂−λ₁/λ₁)), and (V/R₃)cos(2π(D₃−λ₁)/λ₁)+j((V/R₃) sin(2π(D₃−λ₁/λ₁)), respectively, where j isthe square root of minus one. The measured total current I₁ obtained bymaking a current measurement 29 at the downstream electrode 22 afterapplying frequency f₁ will be given by the sum of these three complexvalues.

The analysis of the preceding two paragraphs can be repeated at twoadditional frequencies f₂ and f₃, corresponding to additionalwavelengths λ₂ and λ₃, and giving branch currents in circuit branches23, 24, 25 of (V/R₁) cos(2π(D₁−λ₂/λ₂)+j((V/R₁) sin(2π(D₂−λ₂/λ₂)), (V/R₂)cos(2π(D₂−λ₂/λ₂)+j((V/R₂) sin(2π(D₂−λ₂/λ₂)), and (V/R₃)cos(2π(D₃−λ₂/λ₂)+j((V/R₃) sin(2π(D₃−λ₂/λ₂)), respectively, when thesignal at frequency f₂ is applied, and (V/R₁) cos(2π(D₁−λ₃/λ₃)+j ((V/R₁)sin(2π(D₂−λ₃/λ₃)), (V/R₂) cos(2π(D₂−λ₃/λ₃)+j((V/R₂) sin(2π(D₂−λ₃/λ₃)),and (V/R₃) cos(2π(D₃−λ₃/λ₃)+j((V/R₃) sin(2π(D₃−λ₃/λ₃)), respectively,when the signal at frequency f₃ is applied. Again, the branch currents,expressed in complex form, may be summed to give the currents I₂ and I₃that will be measured at the downstream electrode 22.

The relations described in the three preceding paragraphs comprise alinear system as follows:

${\frac{I_{1}}{V} = {{\left( \frac{1}{R_{1}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{1} - \lambda_{1}} \right)}{\lambda_{1}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{1} - \lambda_{1}} \right)}{\lambda_{1}}} \right)}}} \right)} + {\left( \frac{1}{R_{2}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{2} - \lambda_{1}} \right)}{\lambda_{1}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{2} - \lambda_{1}} \right)}{\lambda_{1}}} \right)}}} \right)} + {\left( \frac{1}{R_{3}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{3} - \lambda_{1}} \right)}{\lambda_{1}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{3} - \lambda_{1}} \right)}{\lambda_{1}}} \right)}}} \right)}}},{\frac{I_{2}}{V} = {{\left( \frac{1}{R_{1}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{1} - \lambda_{2}} \right)}{\lambda_{1}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{1} - \lambda_{2}} \right)}{\lambda_{2}}} \right)}}} \right)} + {\left( \frac{1}{R_{2}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{2} - \lambda_{2}} \right)}{\lambda_{2}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{2} - \lambda_{2}} \right)}{\lambda_{2}}} \right)}}} \right)} + {\left( \frac{1}{R_{3}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{3} - \lambda_{2}} \right)}{\lambda_{2}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{3} - \lambda_{2}} \right)}{\lambda_{2}}} \right)}}} \right)}}},{\frac{I_{3}}{V} = {{\left( \frac{1}{R_{1}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{1} - \lambda_{3}} \right)}{\lambda_{3}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{1} - \lambda_{3}} \right)}{\lambda_{3}}} \right)}}} \right)} + {\left( \frac{1}{R_{2}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{2} - \lambda_{3}} \right)}{\lambda_{3}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{2} - \lambda_{3}} \right)}{\lambda_{3}}} \right)}}} \right)} + {\left( \frac{1}{R_{3}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{3} - \lambda_{3}} \right)}{\lambda_{3}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{3} - \lambda_{3}} \right)}{\lambda_{3}}} \right)}}} \right)}}}$

The quantities 1/R₁, 1/R₂, 1/R₃ may therefore be determined by solvingthe foregoing linear system, using any of the many techniques forsolving linear systems that are known to persons having ordinary skillin the art of linear mathematics, such as, by way of example only,Gaussian elimination or matrix inversion. To do this it is merelynecessary to know V, which in the case of actual physical measurement isknown because it is the fixed voltage applied; I₁, I₂, and I₃, which areknown by measuring the current amplitudes and phases directly using asuitable instrument 29 at the downstream electrode 22 upon applyingfrequencies corresponding to wavelengths λ₁, λ₂, and λ₃; the pathlengths D₁, D₂, and D₃, which in this example are given and in thecontext of an actual measurement on an object would be determined bydirect measurement of the straight line distance D₁ between electrodesand choosing suitable arbitrary values for D₂ and D₃ as described infra;and the wavelengths λ₁, λ₂, and λ₃, which may be determined from theknown frequencies applied, assuming the speed of light in the medium isknown (and if not it can be measured or estimated).

The linear system can be expressed in matrix form and generalized to anyarbitrary number of resistance paths and interrogating signalwavelengths, as follows:

$\left( \frac{I_{n}}{V_{n}} \right) = {\left( {{\cos\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)}}} \right)\left( \frac{1}{R_{m}} \right)}$where

$\left( \frac{I_{n}}{V_{n}} \right)$is a column vector of n observed currents (scaled by the applied voltageamplitude, which can optionally be different for each frequency, hencethe subscript), expressed in polar form, at the downstream electrodeupon applying an alternating current at potential V and frequencies f₁ .. . f_(n) corresponding to wavelengths in the medium being interrogatedof λ₁ . . . λ_(n);

$\left( \frac{1}{R_{m}} \right)$is a column vector of the m resistances in paths 1 . . . m; and

$\left( {{\cos\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)}}} \right)$is a n (rows) by in (columns) matrix of coefficients corresponding tothe phase shift attributable to the difference between the path lengthalong path m and the wavelength of the applied signal. The number ofdifferent interrogating frequencies n must be chosen to be at leastequal to the number of paths to be evaluated m; if the number offrequencies n is less than the number of paths m, the system will ingeneral be underdetermined, and if the number of frequencies n exceedsthe number of paths m, the system will in general be overdetermined. Inthe latter case, use may be made of various techniques known to personshaving ordinary skill in the art of linear mathematics for improving theaccuracy of the solution by utilizing the additional informationembodied in the constraints exceeding the number of degrees of freedomof the system.Extending the Model to Account for Capacitance

The foregoing system and method can be extended to account forcapacitance effects by including capacitance branches, as shown withrespect to the three-branch example in FIG. 2. Capacitances of the threebranches are represented by three additional parallel paths 31, 32, 33having capacitances, respectively, of C₁, C₂, and C₃ and having lengthsD₁, D₂, and D₃, respectively, identical to those of the correspondingresistance paths. In effect, each of the three paths is treated as aparallel RC path, and represented as two separate paths, one a pureresistance path, and one a pure capacitance path. The current due to asinusoidal voltage signal through a capacitance is inverselyproportional to the capacitive reactance,

${\frac{1}{2\pi\;{fC}} = \frac{\lambda}{2\pi\;{cC}}},$where f is the frequency of the signal, C is the capacitance, λ is thewavelength of the signal, and c is the speed of light in the mediumthrough which the signal is passing. According, the expected pathcurrent I at the downstream electrode for a path with capacitance C,path length D, at interrogation wavelength λ is given by

${I\left( {\lambda,D,C} \right)} = {{V \cdot {jC} \cdot \left( \frac{2\pi\; c}{\lambda} \right)}\left( {{\cos\left( {2\pi\frac{\left( {D - \lambda} \right)}{\lambda}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D - \lambda} \right)}{\lambda}} \right)}}} \right)}$The coefficient,

${\left( \frac{2\pi\; c}{\lambda} \right)\left( {{\cos\left( {2\pi\frac{\left( {D - \lambda} \right)}{\lambda}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D - \lambda} \right)}{\lambda}} \right)}}} \right)},$depends only on D and λ (making the assumption that c is known or can beestimated or measured and that the medium is isotropic regarding c alongthe path in question). Therefore, the linear system describing thecurrents measured at the downstream electrode 22 (see FIG. 2) is givenby:

$\left( \frac{I_{n}}{V_{n}} \right) = {\left\{ {\left( {{\cos\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)} + {{j \cdot \sin}\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)}} \right),{\left( \frac{2\pi\; c}{\lambda} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)}}} \right)}} \right\}\left\{ {\begin{matrix}\frac{1}{R_{m}} \\{jC}_{m}\end{matrix},} \right\}}$Where

$\left( \frac{I_{n}}{V_{n}} \right)$is again a vector of measured currents upon applying sinusoidal voltagesignals of amplitude V₁ . . . V_(n) and wavelengths λ₁ . . . λ_(n)between electrodes 21 and 22 (such currents expressed in complex form);the matrix of coefficients

$\left\{ {\left( {{\cos\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)} + {{j \cdot \sin}\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)}} \right),{\left( \frac{2\pi\; c}{\lambda_{n}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)}}} \right)}} \right\}$is a n (rows) by 2m (columns) matrix of coefficients, in alternatingcolumns as shown, with the odd numbered columns corresponding toresistance paths and determined according to the expression

$\left( {{\cos\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)} + {{j \cdot \sin}\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)}} \right),$and the even numbered columns corresponding to capacitance paths anddetermined according to the expression

${\left( \frac{2\pi\; c}{\lambda_{n}} \right)\left( {{\cos\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)} + {j \cdot {\sin\left( {2\pi\frac{\left( {D_{m} - \lambda_{n}} \right)}{\lambda_{n}}} \right)}}} \right)};{{and}\mspace{14mu}\left\{ {\begin{matrix}\frac{1}{R_{m}} \\{jC}_{m}\end{matrix},} \right\}}$is a vector of unknown values of 1/R_(m) and jC_(m) to be solved for,with the odd numbered rows representing I/R_(m) values and the evennumbered rows representing jC_(m) values. The C_(m) values (or,alternatively, the matrix coefficients for the even numbered columns)are multiplied by j so as to account for the 90 degree phase shiftcaused by the capacitance. A Mathematica program implementing theforegoing methods and illustrating the 3-resistance path, 3-capacitancepath example using specific numeric values follows as Appendix A. Thevalues shown therein are not intended to be physiologically realistic,but merely to illustrate the application of the method and show how itmay be implemented in software.

Inductance is usually thought to be negligible in physiological media,and has not been accounted for in the foregoing example. It will beapparent, however, that the model is easily extended to account forinductance by including additional paths for the inductances in the samemanner as has been done for the capacitances, and including appropriatecoefficients taking into account inductive reactance and the 90 degreephase shift caused by inductance in the opposite direction from thephase shift caused by capacitance. Doing so would obviously necessitateinterrogating at a number of additional frequencies at least equal tothe number of inductance paths added.

The model described above and illustrated in the example can begeneralized to any arbitrary number of paths, limited only by the numberof interrogation frequencies applied, the computational resourcesavailable to handle the large matrices that result from large numbers ofpaths, and the ability to measure currents with sufficient accuracy.

Application of System and Method as to Continuous Medium

Obviously, the foregoing example, dealing as it does with a finitenumber of discrete paths, differs from the type of measurement desiredfor imaging purposes, where the conductive properties of the objectbeing measured are continuous. To understand how the system and methodof the invention may be used for estimating the impedance of acontinuous two or three dimensional object along a straight line pathbetween two electrodes, it will be convenient to consider theconfiguration shown in FIG. 3. Any predetermined group of paths 37(referred to herein as ‘component paths’) may in principle berepresented in the model by a single path 38 (referred to herein as a‘lumped path’). If this is done, the lumped path values determined bythe model will always underestimate the aggregate of the composite pathcurrents, and therefore overestimate the impedance. This is so becausethe component paths are of unequal lengths. Therefore the currents onsuch component paths are delayed relative to the interrogation signal byvarying time intervals in their arrival at the downstream electrode 22,so they are out of phase with each other to varying degrees. The ‘true’aggregate current representing the true impedances of the componentpaths is the in-phase sum of all the component path currents representedby the ‘lumped’ path; to the extent that any of the component pathcurrents is out of phase with the others, the aggregate current must bereduced from the ‘true’ aggregate current when the component pathcurrents are superposed.

The extent to which the ‘lumped’ shunt path current underestimates the‘true’ aggregate of the currents on the paths represented by the‘lumped’ shunt path depends upon the distribution of the path lengths ofsuch paths. In general, the narrower the distribution of path lengths ofthe component paths 37, so that the variation in path length among thecomponent paths is relatively small, then the smaller will be thedifference between the ‘true’ aggregate current and the lumped pathcurrent obtained by the model. If the distribution of the componentpaths is known or can be estimated, then it is possible to compute acorrection factor for the average expected error based on theattenuation to be expected given the distribution of paths present, theinterrogation frequency, and an assumed distribution of currentamplitudes among the component paths.

For imaging purposes, therefore, a somewhat crude measure of thestraight line path impedance can be obtained using the analysisdescribed herein with only two paths: the straight line path for whichthe path length D₁ is measured directly as the distance betweenelectrodes; and a single shunt path, whose path length D₂ is determinedarbitrarily as a reasonable ‘lumped’ shunt path based on the knowngeometry of the object being imaged. For example, for an object that isapproximately symmetrical about the straight line path, any pathcoplanar with the straight line path and equidistant between thestraight line path and the boundary of the object might reasonably betaken as the lumped shunt path. A minimum of four interrogation signalsat different frequencies is applied to the electrodes in the manneralready described, and, for each frequency, the current amplitude andphase is measured at the downstream electrode and, for convenience,converted to a complex value. The linear system described herein issolved and the straight line path resistance and capacitance arecomputed. The procedure is repeated for a plurality of other electrodepair locations. From the resistance (and optionally capacitance) valuesobtained for the various straight line paths, an image may beconstructed by any of the well known methods for constructing an imagefrom line integrals, such as back projection or Fourier analysis.

A Preferred Embodiment

Because the expected underestimation of currents depends upon the rangeof path lengths represented by the lumped paths, it is desirable toreduce the range of path lengths of the component paths corresponding toeach lumped path as much as possible. Therefore, a preferred embodimentof the system and method of the invention proceeds, in general, asfollows: First, a plurality of electrode pair placements is determined,in such a way that the straight line paths between the chosen electrodepair positions are spatially distributed in a manner suitable for imagereconstruction by back projection when the impedances of such straightline paths have been determined, at the desired resolution.

Then, for each electrode position pair so determined, the impedanceproperties of the paths between the two electrodes are determined usingthe methods described herein, and in general as described in thisparagraph. First, unless it is known a priori, the distance D₁ betweenthe two electrodes is measured using a caliper. (As used herein,“caliper” shall mean any apparatus now known or existing at any futuretime, and any equivalent thereof, whose function is to measure thestraight line distance between two points in space.) A distance D_(MAX),being the longest current path to be analyzed, is also determined bymeasuring or estimating the longest distance between the two electrodesalong the circumference of the object being imaged, and coplanar withthe straight line path. (It is recognized that, in theory, current cantake convoluted paths that could be much longer than a directcircumferential path, but it is believed that as a practical matter thecurrents attributable to extremely long paths may reasonably beneglected for imaging purposes. The determination of D_(MAX) can be madein many possible ways; the method described is merely one preferredmethod.) It is then necessary to determine the number of shunt paths tobe analyzed, which also determines the number of interrogationfrequencies to be applied. In this preferred embodiment, approximately100 shunt paths is considered a suitable number; taking into accountcapacitance paths, this results in a linear system of rank approximately200, which is readily and quickly solvable on a typical personalcomputer using appropriate software. The lengths of the assumed shuntpaths are distributed evenly over the range from D₁ and D_(MAX),inclusive. Interrogation frequencies are then selected. The number offrequencies must at least equal the number of path lengths; it isrecommended that a larger number be used, in case any readings must bediscarded on account of error or in case it is desired to select thosefrequencies that best optimize the conditioning of the coefficientmatrix. Selection of the frequency range depends upon several competingconsiderations. Interrogation signals of shorter wavelengths aredesirable from the standpoint of giving larger relative phase shiftsbetween paths, making the system less sensitive to measurement error.However, very short wavelengths (i.e. in the microwave range) do notpropagate well in tissue, cause local heating effects, and may causeartifacts if the difference between the lengths of any two paths isgreater than half the wavelength, since in that case the two currentsshift in phase with respect to each other enough to return to more orless in-phase alignment. It is believed that for purposes ofphysiological imaging, interrogation signal wavelengths ranging from onthe order of a minimum (λ_(MIN)) approximately three or four timesD_(MAX) to a maximum (λ_(MAX)) of approximately 20 times D_(MAX)represent an appropriate compromise for this preferred embodiment. Theinvention is not, however, limited to such frequencies, and, as isapparent from the discussion above, in principle any frequencies can beused. The desired number of interrogation wavelengths may be distributedevenly over the range from λ_(MIN) to λ_(MAX); it may also be deemeddesirable to then adjust each of these evenly distributed wavelengths bya small random factor so as to avoid wavelengths that are even multiplesof other wavelengths. This adjustment is believed to possibly improvethe conditioning of the coefficient matrix. A steady state sinusoidalsignal of constant voltage amplitude at each interrogation frequency isapplied in turn at one of the electrodes, and the amplitude and phase ofthe current is measured at the other electrode, in the general mannershown in FIG. 2 and described above. These current amplitudes and phasesare converted to complex values, those values are divided by the voltageamplitude of the applied signal, and the resulting complex values areassembled into the vector of observed currents.

$\left( \frac{I_{n}}{V_{n}} \right).$(In this preferred embodiment, the same voltage amplitude is used forall interrogation signals; however, it is possible to use a differentvoltage for each.) From the distribution of path lengths and thedistribution of interrogation signal wavelengths, the coefficient matrixis assembled as described above. The linear system is then solved forthe resistances and capacitances of each of the approximately 200 paths(100 resistance paths and 100 capacitance paths). The straight line pathresistance and capacitance are noted for the electrode pair position inquestion. (In this preferred embodiment, the other values are not used.)

Having thus determined values for the resistance and capacitance of thestraight line path for all of the selected electrode position pairs, animage is then constructed by back projection or by Fourier analysis.

Another Preferred Embodiment (Finite Element Approach)

In another preferred embodiment, a finite element representation of theobject to be imaged is first determined, having a predetermined numberof nodes at selected positions on the exterior of the object, anadditional predetermined number of nodes at selected positions in theinterior of the object, and edges joining neighboring nodes. Thedetermination of the number and positions of nodes and the choice ofnode pairs to be connected by edges is accomplished in accordance withmethods that are well known to persons having ordinary skill in the artof finite element analysis of the electrical properties of two and threedimensional objects. Pairs of exterior nodes, preferably on generallyopposite sides of the object, are selected for application of theinterrogation signals, thus establishing a set of electrode positionpairs. The number of pairs to be included in the set is at leastsufficient to determine a linear system as described below.

For each electrode position pair, a set of path impedances is determinedas follows: First, the desired number of interrogation frequencies isdetermined. This should be at least equal to twice the number of edgesin the finite element grid, and optimally a larger number should be usedin case any readings must be discarded on account of error, in case itis desired to select those frequencies that best optimize theconditioning of the coefficient matrix, or in case the geometry of thefinite element grid turns out to be such that some edges are notadequately interrogated by fewer frequencies. Then the number of pathsto be analyzed is determined; this should be at least twice the numberof edges in the finite element grid. Next, the possible paths throughthe grid between the two electrode nodes are enumerated in order by pathlength. For smaller grids, this may be done by enumerating all possiblepaths and sorting them by length; for larger grids, the number ofpossible paths makes this impracticable, and Monte Carlo methods may beused, or methods may be applied to generate the possible paths inascending order by length, where the grid topology lends itself to suchmethods. The result of this analysis will be a list of at least as manypaths, ordered by path length, as required to produce a number of pathlengths equal to the number of paths to be analyzed as previouslydetermined. A pair of electrodes is then placed on the object at thepredetermined positions, the interrogation signals are applied, and theamplitudes and phases of the resulting currents are recorded as before.A coefficient matrix is constructed as before from the predeterminedpath lengths and interrogation signal wavelengths, and the linear systemis solved for the resistance and capacitance of each of thepredetermined path lengths.

When the analysis described in the preceding paragraph has beencompleted for each of the predetermined electrode position pairs,another linear system is constructed in which each row represents theequation for one path, as follows:

${R_{PATH} = {\sum\limits_{K}R_{EDGE}}},$where R_(PATH) is the total resistance of the path, R_(EDGE) is theresistance of a single one of the edges comprising the path, and K isthe number of edges comprising the path. This linear system can then besolved for the resistances of the edges. Using the path capacitancesdetermined for all the paths, the capacitances of the edges can besimilarly determined, keeping in mind that serial capacitances combineas

$\frac{1}{C_{PATH}} = {\sum\limits_{K}{\frac{1}{C_{EDGE}}.}}$It may be necessary to adjust the number of paths included in thesystem, and to determine which paths should be used assuming data hasbeen taken for a sufficient number of paths to overdetermine the system,in such a way as to optimize the conditioning of the system and allowdetermination of values for all edges. Since this approach results inresistance and/or capacitance values for all edges in the finite elementgrid, it in effect directly produces what amounts to an image of theresistance or capacitance properties of the object.Interrogation of Straight Line Path Using Resonant Frequency Signal

A further enhancement of the system and method described herein forestimating the magnitude of the current following a straight line pathbetween two electrodes in its passage through a medium having multiplepossible current paths involves the application of an interrogationsignal of such a frequency that resonance is induced along the straightline path. It is well known that when an electrical signal is applied toa conductor at a frequency such that the length of the conductor isequal to or closely approximates one-half the wavelength of the signalin the medium of which the conductor is composed, or an integralmultiple thereof, the conductor will resonate, radiating electromagneticenergy, and thereby causing its observed impedance to increase markedlyat the resonant frequency as compared to its impedance at otherfrequencies. The quantity of energy radiated as electromagnetic energydepends, among other things, upon the amplitude of the signal applied tothe conductor, which in turn is attenuated to the extent it encountersimpedance in its passage through the conductor.

Consider again the model previously described, in which the impedanceproperties of an object are represented as a resistance on a straightline path and one or more shunt resistances and/or capacitances on alonger path. As shown in FIG. 4, a signal generator 38 is used to applya signal whose frequency is such that the half-wavelength of the signalis equal to the sum of the path length along the straight line path andthe length of the conductor through which the signal is applied. Thecurrent will now be split between the straight line path 23 and theshunt path 24 in proportion to the respective impedances. However, theapparent impedance along the straight line path 23 will be greatlyincreased in comparison to the impedance that would be presented to asignal at a non-resonant frequency, because of the energy lost byelectromagnetic radiation, while the apparent impedance along the shuntpath 24 will not be increased. The magnitude of the radiation emittedfrom the resonant signal along the straight line path will depend inpart upon the resistance and impedance properties of the medium of whichthe straight line 23 path is composed, since a more resistive mediumwill attenuate the amplitude of the signal as it passes down theconductor to a higher degree than a less resistive medium would.Therefore, the effect of the impedance properties of the conductivemedium along the straight line path 23 will be relatively magnified interms of their effect upon the observed impedance at resonance. It willbe possible to improve the resolution of the determination of impedancesusing the system and method of the invention, by selecting interrogationfrequencies each of which corresponds to a half-wavelength of which thestraight line path or one of the predetermined shunt paths is anintegral multiple, so that each path used in the analysis isinterrogated by a signal that is resonant along such path.

It will also be possible to use this resonance effect directly toprovide an imageable measure related to the impedance along the straightline path by measuring the amplitude of the electromagnetic fieldradiated at resonance. The amplitude of the electromagnetic fieldradiated from the resonating signal along the straight line path will beinversely related to the impedance properties of the medium comprisingthe path. In a preferred embodiment taking advantage of this effect, aplurality of electrode pair placements is first determined, in such away that the plurality of straight line paths between the chosenelectrode pair positions are spatially distributed in a manner suitablefor image reconstruction by back projection when the impedances of suchstraight line paths have been determined. Then, for each electrodeposition pair so determined, electrodes 21, 22 are placed as illustratedin FIG. 4 and a signal generator 38 is used to apply an interrogationsignal of predetermined amplitude and having a half-wavelength chosen soas to induce resonance in the straight line path 23 between electrodesthrough the object being measured. (This may be accomplished by applyinga signal of whose half-wavelength the straight line path is an integralmultiple, or, using a longer wavelength signal, by inducing resonance inthe straight line path together with a conductor of predetermined lengthby which the signal is applied, as illustrated in FIG. 4. It will bepossible to determine the correct frequency by beginning at a wavelengththat is longer than that required to induce resonance along the straightline path; such a wavelength will instead induce resonance along somelonger path through the object. The frequency will then be graduallyincreased while observing the intensity of the radiated signal. Sincethe straight line path is the shortest possible path, when the frequencyis raised to a point such that the wavelength is slightly less than thatrequired to induce resonance in the straight line path, no path willresonate, and the intensity of the radiated signal will abruptlydecrease. In this way, the frequency required to induce resonance in thestraight line path can be determined.) The intensity of theelectromagnetic radiation so induced will then be measured using a fieldstrength meter 37 or any other of the many methods known to personshaving ordinary skill in the art of detecting and quantifyingelectromagnetic radiation. When such measurements have been made foreach of the predetermined electrode position pairs, the set of measuredintensities, each of which represents a measure of the impedance of thestraight line path between the electrode pair to which it corresponds,will be used to generate an image of the impedance properties of theobject by any of the well known methods for generating images from lineintegrals of physical properties of an object, such as back projectionor Fourier analysis.

Method and Apparatus for Generating Images; Preferred Embodiment

Based on the system and methods described herein, an apparatus can beconstructed for producing images. In a preferred embodiment, as shownschematically in FIG. 5, such an apparatus would comprise a signalgenerator 30 capable of generating interrogation signals at the desiredamplitudes and frequencies; a set of electrodes 21 22 and leads; acaliper 41 for measuring the straight line distance between electrodes;an instrument 29 for measuring accurately the amplitude of the currentat the downstream electrode and its phase relationship with theinterrogation signal; an apparatus 39 for reporting the frequency,amplitude and phase of the interrogation signal and the amplitude andphase of the measured current signal and interfacing with a computer soas to report such data in a form useable by the computer; and aprogrammable computer 40 programmed to carry out the analysis describedherein, to generate an image from the data so provided, and to displaysuch image on a suitable display device 43 such as a monitor or printer.

The computations to be performed in connection with this invention may,of course, be incorporated in software and implemented on the hardwareof a computer programmed in accordance with the methods describedherein. The invention is intended to extend to the apparatus comprisinga computer programmed to carry out the method of the invention, and tomachine readable media upon which has been written or recorded acomputer program for carrying out the method of this invention. Themethods and apparatus of the invention may also be incorporated as partof an imaging apparatus for producing images representing the impedanceproperties of a sample by analyzing measurements taken between aplurality of electrode pairs using existing methods of producing animage from line integral measurements. Such methods are well known inthe field of computed tomography. The invention is intended to extend toany such imaging apparatus incorporating the methods of the invention.

CONCLUSIONS, RAMIFICATIONS, AND SCOPE

The present invention is not limited in scope to the preferredembodiment or examples disclosed herein, which are intended asillustrations of one or a few aspects of the invention, and any methodsthat are functionally equivalent are within the scope of the invention.Various other modifications of the invention in addition to those shownand described herein will become apparent to those skilled in the artfrom the foregoing description. Such modifications are intended to fallwithin the scope of the invention.

In particular, it will be noted that the methods described do notnecessarily require that the path whose impedance properties aremeasured be a straight line path, as long as the path is the shortestconductive path between the electrodes. The number of paths selected foranalysis is not limited to the number described in the examples orpreferred embodiments herein, and is limited only by the computationalresources available and the ability to make accurate current amplitudeand phase measurements for a sufficient distribution of interrogationfrequencies; the larger the number of paths, the better the resolutionobtained, other factors being equal. The selection of interrogationfrequencies is not limited to the ranges described in the examples andpreferred embodiments, and in principle any interrogation frequenciesmay be used provided that the wavelengths are not so short that theattenuation becomes too great for accurate measurement of currents. Theinterrogation signals are not necessarily limited to sinusoidal signals.Non-sinusoidal signals can be decomposed by Fourier analysis into asuperposition of sinusoidal signals at specific frequencies in a mannerwell known to persons having ordinary skill in the art of signalprocessing. Such sinusoidal signals resulting from such decompositioncan then be used to analyze the impedance properties of the object beingimaged using the system and method described herein. It is thereforepossible to apply a non-sinusoidal interrogation signal comprising asuperposition of sinusoidal signals at two or more frequencies, measurethe resulting current signal, decompose the current signal by Fouriermethods into its component sinusoidal signals, and perform the analysisof impedance properties, using the system and method described herein,separately for each of the component sinusoidal signals. Although theinterrogation signal is described in the examples and preferredembodiments herein as having a predetermined, fixed voltage amplitude,obviously the analysis can be carried out using a signal of fixedcurrent amplitude and measuring the voltage signal produced, as will beapparent to a person having ordinary skill in the art of analyzingelectrical signals. Although the examples and preferred embodimentsdiscussed herein reflect the use of only two electrodes, it is possibleto use an array of electrodes interrogated two at a time, provided thatthe straight line distance between electrode pairs is known or can bedetermined. Although the examples and preferred embodiments discussedherein reflect the supposition that the dimensions of the object, andtherefore the straight line distances between electrodes, remainsconstant throughout the measurements, in the case of physiologicalmeasurements of regions such as the torso the dimensions may change dueto breathing, movement, or for other reasons. It will be apparent uponinspection of the linear system describing the relationship between themeasured currents and the path resistances and capacitances that it ismerely necessary for the dimensions to remain constant during eachsingle interrogation of a single electrode position pair at a singlefrequency, which can be accomplished nearly instantaneously. Althoughthe examples and preferred embodiments discussed herein reflect theplacement of electrodes on the outer surface of the object to bemeasured, the system and method described herein are equally applicableto configurations in which either electrode or both electrodes areplaced at a point in the interior of the object. Doing so may makeadvisable the broadening of the range of assumed shunt path lengths,since the currents produced by the interrogation signals can then takeless direct paths. Such configurations may be useful in geologicalapplications where electrodes are placed in drilled holes, and inphysiological measurement where electrodes are placed in the interior ofthe body by surgery, via a needle, or otherwise.

It is not required that the signals used to interrogate the object beelectrical signals; the system and method described herein is applicableto any type of signal characterized by wave-like propagation through anattenuating medium through which the signal tends to follow paths ofleast resistance, including, without limitation, acoustic signals.

It must be recognized that the measurements described herein may notnecessarily provide exact straight line impedance values due to avariety of factors, not least of which is that a straight line has, bydefinition, infinitessimal thickness and therefore infinite impedance.The system and method discussed herein produces a reasonable measurerelated to the impedance along paths of finite thickness, correspondingto a resolution that can improved to an arbitrary extent by increasingthe number of interrogation frequencies. The values produced by themethods described will, however, bear a relation to physiologicalproperties, including impedance, and/or some composite thereof, andprovide quantities that can usefully be employed to produce images thatfurnish information relevant to assessment of physiological states.Therefore, even to the extent that the values produced are not, strictlyspeaking, precisely accurate straight line impedance values, they can beused to produce images that themselves are useful.

APPENDIX A Mathematica Code Illustrating Computation of Specific Examplewith 3 Resistance and 3 Capacitance Paths Corresponding to FIG. 2

1. Computation

(* this file contains the calculations for a n-path RC network. It isintended to be used with the accompanying file EITnpathfns.nb whichcontains many of the functions used herein, so as to avoid clutter. *)

npaths=3; (* each has an R and a C branch so actual no of paths is twicethis *)

(* path table is generated randomly using pathr fn and output copiedhere so that it stays the same on recalculation*)

path={{{15.3235, 46.2676}, 1}, {{23.4823, 138.293}, 1.29172},

-   -   {{25.0759, 113.762}, 1.42838}}        {{{15.3235, 46.2676}, 1}, {{23.4823, 138.293}, 1.29172},        {{25.0759, 113.762}, 1.42838}}        (* wavelengths of signals to be used for interrogating        (arbitrary) *)        sep=0.09;        lambdas=Table [1+(i−1)*sep, {i, 1, 2*npaths}]        {1, 1.09, 1.18, 1.27, 1.36, 1.45}        (* generate matrix of the path currents at the various        interrogation frequencies. Row corresponds to frequency, column        corresponds to path, in order R, C, in increasing length. *)        iin=Table[ilist[pathcc[path, lambdas[[i]]], 10, lambdas[[i]]],        {i, 1, Length[lambdas]}]

-   {{0.652592, 0.581416 i, −0.110357+0.411305 i,    -   −1.67847−0.450349 i, −0.359088+0.17346 i, −0.621818−1.28726 i},

-   {0.566723−0.323578 i, 0.264482+0.463222 i, 0.168969+0.390896 i,    -   −1.46348+0.632602 i, −0.147829+0.370377 i, −1.2181−0.486181 i},

-   {0.375102−0.534018 i, 0.403198+0.283212 i, 0.352698+0.238652 i,    -   −0.82534+1.21975 i, 0.0979815+0.386565 i, −1.17437+0.297664 i},

-   {0.151952−0.634655 i, 0.445224+0.106598 i, 0.423396+0.045673 i,    -   −0.146759+1.36049 i, 0.282502+0.28147 i, −0.794496+0.79741 i},

-   {−0.0602136−0.649809 i, 0.425688−0.0394458 i, 0.404839−0.13212 i,    -   0.396441+1.21477 i, 0.379054+0.123898 i, −0.32658+0.999139 i},

-   {−0.241549−0.606243 i, 0.372498−0.148417 i, 0.329556−0.26971 i,    -   0.759067+0.927494 i, 0.397041−0.0373057 i, 0.0922298+0.981591        i}}        (* i0 is vector of observed currents at each of the        interrogation frequencies, which is the superposition of the        path currents for all paths *)        i0=Table[iout[iin[[i]]], {i, 1, Length[iin]}]        {−2.11715−0.571424 i, −1.82923+1.04739 i, −0.770728+1.89183 i,    -   0.361819+1.95698 i, 1.21923+1.51694 i, 1.70884+0.897409 i}        (* pull out separate vector of pathlengths from paths *)        plengths=Transpose[pathcc[path, 1]] [[2]]        {1, 1, 1.29172, 1.29172, 1.42838, 1.42838}        (* make rotation matrix—element R[a,b] is the complex unit        vector which when multiplied by a signal shifts the phase of        that signal by the amount of the phase shift that would be        observed for a signal of wavelength a measured at a point at a        distance b from the point of application of the signal. In the        case of the capacitance paths, R[a,b] is scaled by        2*Pi*(200000000/lambda)*(1/1000000000000) to account for the        dependence of capacitive reactance on frequency, the assumed        speed of light in the medium, and to allow capacitances to be        expressed in picofarads, and multiplied by I to account for the        90 degree phase shift on a capacitance path. *)        R=mrot[plengths,lambdas]

-   {{1., 0.00125664 i, −0.259143+0.965839 i,    -   −0.00121371−0.000325648 i, −0.900446+0.434967 i,        −0.000546596−0.00113153 i},

-   {0.868417−0.495834 i, 0.000571636+0.00100118 i, 0.396778+0.917915 i,    -   −0.00105824+0.000457436 i, −0.370696+0.928754 i,        −0.00107074−0.000427367 i},

-   {0.574787−0.813303 i, 0.000871449+0.000612118 i, 0.828216+0.560409    i,    -   −0.000596805+0.000882006 i, 0.245697+0.969347 i;        −0.0010323+0.000261655 i},

-   {0.232844−0.972514 i, 0.000962281+0.000230394 i, 0.994232+0.107251    i,    -   −0.000106122+0.000983771 i, 0.7084+0.705811 i,        −0.000698385+0.000700946 i},

-   {−0.0922684−0.995734 i, 0.000920056−0.0000852558 i,    0.950656−0.310247 i,    -   0.000286668+0.000878404 i, 0.950513+0.310686 i,        −0.000287073+0.000878272 i},

-   {−0.370138−0.928977 i, 0.000805094−0.000320779 i, 0.773872−0.633342    i,    -   0.000548883+0.000670673 i, 0.995615−0.0935475 i,        0.0000810726+0.000862846 i}}        (* now should be able to solve the linear system i=v*R.sigma for        sigmas *)        Chop[LinearSolve[10*R, i0], 10^−7]        {0.0652592, 46.2676, 0.0425853, 138.293, 0.0398789, 113.762}        (* which returned the exact capacitances we started with and the        reciprocals of the resistances. *)        2. Mathematica Functions Required for Computation in (1) Above        (* each path is a {R,C} and a length in that order, here sort of        randomly chosen but in increasing lengths. We will use base freq        of 200 MHz corresponding to wavelength of 1 M at a c of 2 E8        m/sec, and resistances an the order of 10 ohms, capacitances on        the order of 100 pf, giving currents on the order of 1 A *)        pathr[n_]:=    -   Module[{i, p},        -   p=            -   Table[{{Random[Real, {(i−1)*10, (i+1)*10}], Random[Real,                {(i−1)*50, (i+1)*50}]},                -   Random[Real, (0.9+(i−1)*0.2, 1.1+(i−1)*0.2}]}, (i,                    1, n}];        -   p[[1,2]]=1;        -   p    -   ]        pathr[3]        {{{10.5161, 83.9719}, 1}, {{29.6562, 136.392}, 1.13124},        {{25.4662, 123.823}, 1.32228}}        (* convert a signal given as an amplitude and phase to polar        form *)        z[amp_, phase_]:=Module[{x, y},    -   x=amp*Cos[phase];    -   y=amp*Sin[phase];    -   x+I* y    -   ]        (* path complex conductivity as function of wavelength, path        length, and R and C *)        sigma[r_, c_, lambda_]:=    -   Module[{i}, 1/r+I*(2*Pi* (200000000/lambda)*(c/1000000000000))    -   ]        (* from a path expressed as {{{R ohms, C pf}, pathlength in        meters} . . . } generate a path table that treats the R and C as        separate paths, at a specified reference wavelength lambda, each        element of the new path expressed as {complex conductivity,        length}*)        pathcc[path_, lambda_]:=Module[{pt},    -   pt=Table[{{Re[sigma[path[[i, 1, 1]], path[[i, 1, 2]], lambda]],        path[[i, 2]], {I*Im[sigma[path[[i, 1, 1]], path[[i, 1, 2]],        lambda]], path[[i, 2]]}}, {i, 1, Length[path]}];    -   Flatten[pt, 1]    -   ]        (* given an angle_ in radians, generate a complex number of        phase equal to angle_ and amplitude 1 (which can be multiplied        by any other complex number to rotate its phase by angle_)*)        unitz[angle_]:=Module[{ },    -   imz=Sin[angle];    -   rez=Cos[angle];    -   N[(rez+I*imz)]    -   ]        (* given a path length plen_ and a wavelength lambda_, generate        a unit complex rotation factor corresponding to the phase shift        caused by the path length difference—i.e. applying a signal of        wavelength lambda, what is the phase of that signal at a        distance plen from the point of application *)        zrot[plen_, lambda_]:=Module[{ },    -   Chop[unitz[((plen−lambda)/lambda)*2*Pi]]    -   ]        (* given a set of path lengths and a set of interrogation        wavelengths, generate a rotation matrix of which the rows        correspond to the interrogation wavelengths, the columns        correspond to the path lengths in order R, C, R, C . . . in        increasing length, and each element mrot[a,b] is the complex        unit vector which when multiplied by a signal shifts the phase        of that signal by the amount of the phase shift that would be        observed for a signal of wavelength a measured at a point at a        distance b from the point of application of the signal. *)        mrot[plens_, lambdas_]:=Module[{i, j, temp},    -   temp=Table[Table[zrot[plens[[i]], lambdas[[j]]], {i, 1,        Length[plens]}], {j, 1, Length[lambdas]}];    -   For[j=1, j≦Length[lambdas], j++,        -   For[i=2, i≦Length[plens], i+=2,            -   temp[[j, i]]*=I*2*Pi*                (200000000/lambdas[[j]])*(1/1000000000000); ];];    -   temp    -   ]        (* This returns the phase angle of a signal expressed in polar        form. The problem with arctan is that tan is the same in first        and third quadrants and in second and fourth. Therefore need to        distinguish based on actual sign of both Re and Im. *)        phasefn[zin_]:=Module[{ },    -   If[Re[zin]==0 && Im[zin]>0, Return[Pi/2],];    -   If[Re[zin]==0 && Im[zin]<0, Return[3Pi/2],];    -   If[Re[zin]≧0,        -   If[Im[zin]≧0,            -   ArcTan[Im[zin]/Re[zin]],            -   2*Pi+ArcTan[Im[zin]/Re[zin]]],        -   If[Im[zin]≧0,            -   Pi+ArcTan[Im[zin]/Re[zin]],            -   Pi+ArcTan[Im [zin]/Re [zin]]]        -   ]    -   ]        (* generate current signals in polar form for each path for an        applied signal of voltage v and wavelength lambda, based on path        resistance and length, using a table of paths in the form        generated by fn pathcc. Note there is no need to treat        capacitances differently because the impedances are already        rotated by expressing them as imaginary in the path list created        using pathcc. *)        ilist[path_, v_, lambda_]:=Module[{i},    -   Chop[Table[v*path[[i, 1]]*zrot[path[[i, 2]], lambda], {i, 1,        Length[path]}]]    -   ]        (* resultant current signal of a list of superposed signals *)        iout[ilist_]:=Module[{i},    -   Sum[ilist[[i]], {i, 1, Length[ilist]}]    -   ]

I claim:
 1. A method for estimating a measure of an impedance property between a first locus and a second locus separated by a conductive medium having at least two electrically conductive paths between the first locus and second locus, the method comprising: introducing at the first locus an alternating electrical signal at a frequency at which at least one conductive path resonates; and measuring the intensity of electromagnetic radiation emitted by the at least one conductive path that resonates.
 2. The method of claim 1, wherein a conductor is electrically connected to the second locus, and the at least one conductive path includes all or part of the conductor.
 3. The method of claim 1, further comprising determining the amplitude of the alternating electrical signal.
 4. The method of claim 1, wherein the alternating electrical signal is propagated ionically over at least part of a conductive path.
 5. The method of claim 1, wherein at least one of said conductive paths passes through a body or body part of an organism.
 6. The method of claim 2, wherein the conductor is electrically connected to the second locus via an electrode.
 7. The method of claim 2, wherein the at least one of said conductive paths comprises a shortest-length conductive path from the first locus to the distal end of the conductor.
 8. The method of claim 1, wherein the length of a conductive path from the first locus to the second locus is approximately an integral multiple of a half-wavelength of said alternating electrical signal.
 9. The method of claim 2, wherein the sum of the length of the conductor plus the length of a conductive path from the first locus to the second locus is approximately an integral multiple of a half-wavelength of said alternating electrical signal.
 10. A method of estimating a spatial distribution of a measure of impedance properties comprising: estimating, according to the method of claim 2, a measure of an impedance property between the first and second locus of a plurality of pairs of loci; and computing, from the estimates of the impedance property and the positions of the loci, an estimate of a spatial distribution of the measure of impedance properties.
 11. An apparatus for estimating a measure of an impedance property between a first locus and a second locus, there being at least two electrically conductive paths between the first locus and second locus, the apparatus comprising: a signal generator operable to generate an alternating electrical signal; a first electrode for electrically connecting the output of the signal generator to the first locus; a conductor; a second electrode for electrically connecting the conductor to the second locus; and a detector disposed to measure the intensity of the electromagnetic radiation emitted by the conductor.
 12. The apparatus of claim 11, further comprising a computer programmed to compute, from estimates of a measure of an impedance property between a plurality of pairs of loci, an estimate of a spatial distribution of the impedance property. 